TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Noise-robust existence of generalized FNO approximants for EIT

Statement

Assume the Axiom of Choice and Countable Choice. Let $d \geq 2$ , let $Ω \subset (0, 1)^{d}$ be nonempty, bounded, connected and $C^{\infty}$ , let $Ω^{'} ⋐ Ω$ , and fix the ellipticity, collar and regularity parameters defining $X_{d} (R)$ . Let $M = \partial Ω$ , let $K \subset L^{2} (M \times M; C)$ be compact, and for $δ \geq 0$ put $K_{δ} = {η \in K : ∥ η ∥_{L^{2} (M \times M; C)} \leq δ}$ . Let $κ_{γ}$ be the zero-mean-in-each-variable kernel of the dimension-dependently smoothed mean-zero NtD observation for $γ \in X_{d} (R)$ . Regard $L^{2} (M \times M; C)$ as a real Hilbert space and let $R f = (Re f, Im f)$ be its isometric realification into $L^{2} (M \times M; R^{2})$ . Let ${\tilde{Ψ}}^{⋆} : L^{2} (M \times M; C) \to L^{2} (Ω; R)$ extend the exact-data inverse map and satisfy $∥ {\tilde{Ψ}}^{⋆} (f) - {\tilde{Ψ}}^{⋆} (g) ∥_{L^{2} (Ω; R)} \leq C_{⋆} ω_{⋆} (∥ f - g ∥_{L^{2} (M \times M; C)}),$ where $ω_{⋆}$ is continuous, nondecreasing and concave with $ω_{⋆} (0) = 0$ . Then for every $ε > 0$ there exists a generalized Fourier neural operator $Φ : L^{2} (M \times M; R^{2}) \to L^{2} (Ω; R)$ with two real input channels such that, on complex inputs, $Ψ (f) : = Φ (R f)$ satisfies $\sup_{γ \in X_{d} (R), η \in K_{δ}} ∥ γ - Ψ (κ_{γ} + η) ∥_{L^{2} (Ω; R)} \leq ε + C_{⋆} ω_{⋆} (δ) .$ This is an existence statement for continuum NtD kernels and compact noise sets; it makes no training, finite-sample, network-size, finite-element, electrode-matrix or calibration guarantee.

Statement references

The dimension-dependent smoothed mean-zero NtD kernel convention.

The Hilbert–Schmidt kernel representation and kernel stability interface.

Compactness of the exact kernel image.

The stable Hilbert-space extension of the exact inverse map.

Generalized FNO approximation on compact input sets.

de Hoop et al., Theorem 4.8 and proof

Facts & Assumptions

Given: the EIT domain and class, compact complex-valued perturbation set $K$ , noise threshold $δ$ , kernel map and Hilbert extension in the Statement.

[F1]

The exact kernel image $K_{X} = {κ_{γ} : γ \in X_{d} (R)}$ is compact in $L^{2} (M \times M)$ . (EIT forward continuity and compactness of conductivity classes)

[F2]

The extension agrees with the exact-data inverse, ${\tilde{Ψ}}^{⋆} (κ_{γ}) = γ$ , and satisfies the displayed $C_{⋆} ω_{⋆}$ bound. (Stable extension of the EIT inverse map, Hilbert-valued extension with a concave modulus)

[F3]

$K_{δ}$ is a closed subset of compact $K$ , and addition is continuous on the complex $L^{2}$ space viewed as a real Hilbert space.

[F4]

A continuous map from $L^{2} (M \times M; R^{2})$ to $L^{2} (Ω; R)$ can be uniformly approximated on every compact input set by a generalized FNO of the stated multichannel architecture. (Generalized FNO approximation on compact sets)

[A1]

The Axiom of Choice and Countable Choice are used at the conductivity, boundary spectral, and Hilbert extension interfaces. (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

Given: the data in the Statement.

1.1A1F1F3givenconstruct

If $K_{δ} = \emptyset$ or $X_{d} (R) = \emptyset$ , choose the zero generalized FNO; the uniform assertion is then vacuous. Otherwise, [A1] applies at the upstream conductivity, spectral and extension interfaces. [F1] and compactness of $K_{δ}$ make $Z_{δ} : = K_{X} + K_{δ} = {κ + η : κ \in K_{X}, η \in K_{δ}}$ compact in complex $L^{2}$ , since it is the continuous image of $K_{X} \times K_{δ}$ under addition. The isometric realification $R (Z_{δ})$ is therefore compact in $L^{2} (M \times M; R^{2})$ .

2.1F4step 1.1choose

Apply [F4] to the continuous real-valued map ${\tilde{Ψ}}^{⋆} \circ R^{- 1}$ on the compact set $R (Z_{δ})$ with tolerance $ε$ . There is a two-input-channel generalized FNO $Φ$ with $\sup_{z \in Z_{δ}} ∥ {\tilde{Ψ}}^{⋆} (z) - Φ (R z) ∥_{L^{2} (Ω; R)} < ε$ .

3.1F2step 2.1givenalgebra∎

For $γ \in X_{d} (R)$ and $η \in K_{δ}$ , use [F2], the triangle inequality and monotonicity of $ω_{⋆}$ to obtain $\begin{aligned} ∥ γ - Φ (R (κ_{γ} + η)) ∥_{2} & \leq ∥ {\tilde{Ψ}}^{⋆} (κ_{γ}) - {\tilde{Ψ}}^{⋆} (κ_{γ} + η) ∥_{2} \\ + ∥ {\tilde{Ψ}}^{⋆} (κ_{γ} + η) - Φ (R (κ_{γ} + η)) ∥_{2} \\ \leq C_{⋆} ω_{⋆} (∥ η ∥_{2}) + ε \\ \leq C_{⋆} ω_{⋆} (δ) + ε . \end{aligned}$ Here ${\tilde{Ψ}}^{⋆} (κ_{γ}) = γ$ because the extension agrees on the exact real kernel image, and the complex $L^{2}$ norm is preserved by $R$ . Taking the supremum gives the claim.

Depends on

Used by

Continuum approximation theorem versus finite-element experiments Example
Finite-sample generalization of learned inverse maps Open problem

Dependency tree · two levels

37 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Maarten V. de Hoop, Nikola B. Kovachki, Matti Lassas and Nicholas H. Nelsen, Extension and neural operator approximation of the electrical impedance tomography inverse map (standard reference, not scraped)